Video Transcript
In the opposite figure, line 𝐴𝐵
is parallel to line 𝐶𝐹, line segment 𝐴𝐷 is parallel to line segment 𝐵𝐶, and
line segment 𝐴𝐹 is parallel to line segment 𝐵𝐸. If the area of parallelogram
𝐴𝐵𝐶𝐷 is eight square centimeters, find the area of parallelogram 𝐴𝐵𝐸𝐹.
We can begin this question by
noting that we are given the information about three pairs of parallel lines and
line segments. Line 𝐴𝐵 is parallel to line
𝐶𝐹. Line segments 𝐴𝐷 and 𝐵𝐶 are
parallel. And line segments 𝐴𝐹 and 𝐵𝐸 are
parallel. This information also confirms that
we do indeed have two parallelograms. 𝐴𝐵𝐶𝐷 has two pairs of opposite
sides parallel. And the same is true for
𝐴𝐵𝐸𝐹. So given that the area of 𝐴𝐵𝐶𝐷
is eight square centimeters, we need to calculate the area of parallelogram
𝐴𝐵𝐸𝐹.
To help us with this, it is
important that we observe a property of these two parallelograms. And that is that they share this
common base of the side 𝐴𝐵. And given this fact, we can apply
the following theorem. Parallelograms between a pair of
parallel lines have the same area when their bases are the same length or when they
share a common base. The reason why this is true simply
arises from the fact that the area of a parallelogram is found by multiplying the
base by the perpendicular height.
Because parallel lines are always
the same distance apart, then two parallelograms between a pair of parallel lines
will always have the same perpendicular height. So if the base is the same length
or in common between the parallelograms, then their areas will be the same. And so the area of parallelogram
𝐴𝐵𝐸𝐹 is equal to the area of parallelogram 𝐴𝐵𝐶𝐷. And given that 𝐴𝐵𝐶𝐷 has an area
of eight square centimeters, then 𝐴𝐵𝐸𝐹 also has the same area. It is eight square centimeters.
